Weak direct product decomposition of algebras. (English) Zbl 0632.08004

General Algebra, Proc. Conf., Salzburg/Austria 1986, Contrib. Gen. Algebra 5, 105-121 (1987).
[For the entire collection see Zbl 0626.00009.]
Let \(A_ i\) (i\(\in I)\) be finitary algebras of the same type. A subalgebra B of the direct product \(\Pi (A_ i: i\in I)\) is said to be a full subdirect product of the algebras \(A_ i\), if (i) B is a subdirect product of \(A_ i\) (i\(\in I)\), and (ii) for each \(i\in I\) and x,y\(\in B\) there is \(z\in B\) such that \(x(i)=z(i)\) and \(y(j)=z(j)\) for each \(j\in I\) with \(j\neq i\). Let A be a congruence distributive algebra having the property that each congruence on A is a join of completely join irreducible congruences. The main results of the paper are as follows. For each system of full subdirect decompositions of A there exists a common refinement. A is a weak direct product of directly indecomposable factors. Each full subdirect decomposition of A into directly indecomposable factors is a weak direct product decomposition of A. Some applications to ternary algebras are given.
Reviewer: J.Jakubík


08A30 Subalgebras, congruence relations
08B10 Congruence modularity, congruence distributivity
08B25 Products, amalgamated products, and other kinds of limits and colimits


Zbl 0626.00009